Somashekhar Naimpally
Proximity and Hyperspace Topologies
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 57, 99-110(2003)
54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
54B20 Hyperspaces
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54D30 Compactness
Abstract: In this paper we give a survey of the use of proximities in
hyperspace topologies. A proximal hypertopology corresponding to a
LO-proximity is a g eneralization of the well known Vietoris
topology. In case we start with an EF-proximity, the proximal
hypertopology equals the Hausdorff uniform topology corresponding
to the totally bounded uniformity and, being contained in both the
Vietoris and Hausdorff uniform topologies, serves as a bridge
between the two. Wattenberg and Beer-Himmelberg-Prickry-Van Vleck
showed that the locally finite hypertopology induced by a
metrizable space is the sup of the Hausdorff metric topologies
induced by all compatible metrics. Naimpally-Sharma showed that
this follows from the fact that a Tychonoff space is normal iff
its fine uniformity induces the locally finite hypertopology. Di
Concilio-Naimpally-Sharma showed that in a Tychonoff space the
fine uniformity induces the proximal locally finite hypertopology.

We study DELTA topologies introduced by Poppe, and their proximal
variations. We show that a short proof can be given of the
Beer-Tamaki result concerning the uniformizability of (proximal)
DELTA hypertopologies via the Attouch-Wets approach used by Beer
in dealing with the Fell topology. Finally we present a result
concerning (Proximal) DELTA-U-hypertopolgies. Several new
hypertopologies are introduced.
Keywords: ap proximity, hyperspace, $\Delta$-topology, proximal $\Delta$-topology, U-topology, $\Delta\text{U}$-topology, proximal $\Delta\text{U}$-topology, Function space, Vietoris topology, Fell topology, Hausdorff uniformity
Notes: Dedicated to my friend Professor Dr. Harry Poppe on his 70$^{th}$ birthday